California is revising its Math Framework. A draft has been updated after a first round of public comments, and a second round will take place starting in December. The Framework is of course an important document that will affect math instruction in public schools throughout the state for at least several years. I have not seen the draft, but in order to see where things stand, I read the Department of Education Instructional Quality Commission’s Mathematics Framework FAQs. (The IQC is the advisory body that is working on the Framework.)
I am not involved in any of this, but as is my wont, I have opinions. I have written about the topics addressed in the FAQs in the past — I will link to relevant posts, and to relevant articles on my website. It is unlikely that anyone involved in this process cares about my views, but hey, I’m a citizen and taxpayer, I have taught K-12 math for 42 years and I have developed much curriculum. I might as well share my thoughts.
I support the goal of the revision (“improve math outcomes for all students”.) You’d think everyone would agree with this goal, and of course everyone claims to agree. But apparently in the feedback that followed the first draft, there was some anxiety that aiming to help all students could only be done at the expense of “high achievers and gifted students”. This concern is a complicated mix of a sociological phenomenon and educational apprehension. Unfortunately, it is not easily addressed.
Math education has been a tool in maintaining inequality, and those who have benefited from that state of affairs worry about losing that advantage. The way this has worked historically is by a combination of tracking and acceleration.
Tracking is the practice of offering some students an “honors” curriculum with high expectations, and offering other students remedial or low-expectations alternatives. Almost everyone believes that tracking helps everyone: it allows the stronger students to get a solid math education, and it helps the others to get a better grade in an easier course especially designed to help them. Unfortunately, while tracking works reasonably well for the honors track, it has been disastrous for everyone else, resulting in dead-end courses and keeping low-track students out of more advanced math and science courses. This is why NCTM recommends de-tracking in their excellent Catalyzing Change in High School Mathematics.
I support de-tracking, but I expect it will face tremendous resistance. Some of the resistance will come from parents who fear that it will undermine their child’s education and college prospects. And some of the resistance will come from teachers who are justifiably worried about their ability to teach much more heterogeneous classes effectively. The IQC’s response is to encourage “the use of open, authentic, multi-dimensional tasks” and multiple representations. I agree with this recommendation, but I suspect that better materials will not be sufficient. It is entirely possible to teach good materials poorly.
Therefore, another dimension of de-tracking has to involve pedagogy: teachers need to be trained in diversifying their toolbox. “Listen carefully and then practice” will not work well, even if the curriculum is improved. In order to be successful, direct instruction must be combined with techniques that center and prioritize student intellectual engagement. Among other approaches, the thinking classroom, and complex instruction can help. See also my article For A Tool-Rich Pedagogy and the figure in this blog post.
Improving curriculum and pedagogy are worthy goals, and a necessary part of de-tracking secondary schools. Still, the fact remains: even with excellent teaching, students learn math at different rates. I taught in a largely untracked program, and had to come up with strategies to address that. I share those in this article Reaching the Full Range, a concatenation of several blog posts. I cannot go through the whole thing here, but one key idea is what I call differentiation by time, not content. I propose many ways to extend student exposure to important concepts without harming the students who don’t need the extra time. Those techniques can be used along with any combination of curriculum and pedagogy, and they do not involve a steep learning curve. They are easy to implement and they support constant forward motion and eternal review, my recipe for supporting all students. Alas, two formidable obstacles stand in the way: teachers’ habits and departmental culture. Both reform-minded teachers and traditionalists find it hard to break with the tyranny of the clock and the calendar.
Acceleration is the practice of allowing students to take a given course before their age cohort, with a standard example being Algebra 1 in eighth or even seventh grade. This is often motivated by a desire to “get to” Calculus as soon as possible, which is seen as crucial to admission in selective colleges. In partial acknowledgment of student differences, the IQC and the NCTM accept a certain amount of acceleration, but they correctly warn that racing to Calculus can be counter-productive. (Read this guest post by a Penn mathematician: More Calculus, Less Understanding?)
In my view, acceleration by one year can work for many students, and in fact it is a good idea for all concerned —but we should beware of willy-nilly hyper-acceleration. Many parents lose perspective on what is in the best educational interest of the child. (“If many students are taking Algebra 1 in eighth grade, then by God my son should take it in seventh grade!” This turns into an unwinnable arms race.) I have warned about the dangers of that sort of thinking and presented alternatives in this article: Hyper-Acceleration. See also this guest post by a private school teacher On the Desire to Push Kids Ahead.
The Mathematics Framework FAQs include a diagram that shows three versions of the first two years of what has historically been high school math: “Algebra 1/Geometry”, “Integrated”, and “Investigating and Connecting”. I imagine that the first two are based on the CCSSM. I don’t know what the third one is. It will be interesting to see whether this setup will end up being a kind of stealth tracking, with that third pathway being less challenging and rigorous.
The remaining two or three years of high school are to be filled by a wide range of possible math courses, as recommended by NCTM. There too, it will be interesting to see how things play out. (For example, a “data science” course is listed. I shared some thoughts on the current data mania in this blog post: Freakonomics Radio on Math Curriculum.)
The 1992 Framework had some of the same goals as the 2022 Framework. Alas, it generated tremendous backlash, leading to a nearly-unteachable replacement. Given its alignment with a broad professional consensus, the current attempt has more going for it, but the political environment is probably not very different. It will be crucial to do this right! The stakes are high.